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Next: Wednesday, February 9 Up: Lecture Notes Previous: Friday, February 4

Monday, February 7 and Tuesday, February 8

There wasn't a lecture either day; students did proofs at the
board.  I give some proofs similar to those given at the board.

This is not the final version of these notes; I'm planning to add
more proofs.

all problems are from section 5.2.

#3 A -> B | C |- (A -> B) | (A -> C)

1. A -> B | C premise
Goal: (A -> B) | (A -> C)
| 2. ~(A -> B) premise of subproof
| Goal: A -> C
| | 3. A  premise of subsubproof
| | Goal: C
| | 4. B | C mp lines 1,3
| | Goal: ~B (so that disj. elim. can be applied to get C)
| | | 5. B premise of subsubsubproof
| | | Goal: contradiction
| | | 6. A -> B true conclusion, line 5 (see a proof of this derived rule below)
| | | 7. (A -> B) & ~(A -> B) c.i. lines 2,6
| | 8. ~ B neg. elim. lines 5-7
| | 9. C disj. elim. lines 4,8
| 10. A -> C lines 3-9
11. (A -> B) | (A -> C) disj. intro, lines 2-10.

The rule of true conclusion used above is listed by Grantham.

It has the form 

A

-------

B -> A

and can be proved valid as follows:

1. A (premise)
Goal: B -> A
| 2. B premise of subproof
| 3. A copy of line 1
4. B -> A imp. intro, lines 2-3.

The related rule of false antecedent given by Grantham can be proved
thus:

1. ~A  premise
Goal:  A -> B
| 2. A subpremise
| Goal: B
| 3. B neg. elim. lines 2,1
4. A -> B imp. intro. lines 2-3

The rule can be presented thus:

~A

------------

A -> B

#24 ~A | ~B |- ~(A & B)

1. ~A | ~B premise
Goal: ~(A&B)
| 2. A & B subpremise
| Goal: contradiction
| 3. A c.e. line 2
| 4. B c.e. line 2
| 5. ~~A d.n.i. line 3
| 6. ~B disj. elim. (disjunctive syllogism) lines 1,5
| 7. B & ~B c.i. lines 4,6
8. ~(A&B) neg. intro lines 2-7

#7 (A | B) | C |- A | (B | C)

1.  (A | B) | C  premise
Goal: A | (B | C)
| 2. ~A subpremise
| Goal: B | C
| | 3. ~C
| | Goal: B
| | 4. A | B disj. elim. lines 1,3
| | 5. B disj. elim. lines 2,4
| 6. B | C disj. intro. lines 3-5
7. A | (B | C) disj. intro. lines 2-6



Randall Holmes
2000-05-05